g-Expectations with application to risk measures
Date
2013-03-05
Authors
Offwood, Sonja Carina
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Abstract
Peng introduced a typical ltration consistent nonlinear expectation, called a
g-expectation in [40]. It satis es all properties of the classical mathematical expectation
besides the linearity. Peng's conditional g-expectation is a solution to a
backward stochastic di erential equation (BSDE) within the classical framework of
It^o's calculus, with terminal condition given at some xed time T. In addition, this
g-expectation is uniquely speci ed by a real function g satisfying certain properties.
Many properties of the g-expectation, which will be presented, follow from the speci
cation of this function. Martingales, super- and submartingales have been de ned
in the nonlinear setting of g-expectations. Consequently, a nonlinear Doob-Meyer
decomposition theorem was proved.
Applications of g-expectations in the mathematical nancial world have also
been of great interest. g-Expectations have been applied to the pricing of contingent
claims in the nancial market, as well as to risk measures. Risk measures were
introduced to quantify the riskiness of any nancial position. They also give an indication
as to which positions carry an acceptable amount of risk and which positions
do not. Coherent risk measures and convex risk measures will be examined. These
risk measures were extended into a nonlinear setting using the g-expectation. In
many cases due to intermediate cash
ows, we want to work with a multi-period, dynamic
risk measure. Conditional g-expectations were then used to extend dynamic
risk measures into the nonlinear setting.
The Choquet expectation, introduced by Gustave Choquet, is another nonlinear
expectation. An interesting question which is examined, is whether there are
incidences when the g-expectation and the Choquet expectation coincide.
Description
Programme in Advanced Mathematics of Finance,
University of the Witwatersrand,
Johannesburg.